Layer-finiteness of some groups (Q6572961)

A group \(G\) is layer-finite if the set of elements of any given order is finite, an almost layer-finite group is an extension of a layer-finite group by a finite group. The concept of a layer-finite group first appeared without a name in the work of \textit{S. N. Chernikov} [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 50, 71--74 (1945; Zbl 0061.02905)]. A group \(G\) is called a Shunkov group if for any prime \(p\) and for every finite subgroup \(H\leq G\) any two conjugate elements of order \(p\) from the factor-group \(N_{G}(H)/H\) generate a finite subgroup.\N\NThe main result of this paper is Theorem 2: A locally finite group \(G\) is layer-finite if and only if \(G\) satisfies the condition: the normalizer of any non-trivial finite subgroup of \(G\) is a layer-finite group. Another result proven by the author is Theorem 8: Let \(G\) be a periodic Shunkov group without subgroups of the form \(\mathrm{PSL}_{2}(p)\). If in \(G\) the normalizer of any non-trivial finite subgroup is layer-finite, then the group \(G\) is also layer-finite.